I normally use this trick for rusted in nuts in cars: use a cut off wheel with a rotary tool. It should take you no more than 5 minutes to cut off this nut.
If you have access to a tool such as a Dremel tool or the like, getting a set of cutoff discs will cost no more than USD $5. A hand saw would take too long but also doable. There are also faster cutting tools such as air tools that all car shops should have.
Cut carefully not to damage the other parts. You don't have to cut through the entire nut to get it out. Cut enough to weaken it, then use a chisel or a sharp flat screw driver to knock it out or use pliers to pull it out. You don't have to cut right on center either. you can even cut a quarter angle, remove that part, which would sufficiently weaken the nut for removal. The idea is to weaken it enough that it no longer has integrity. There is no need to fully destroy it in order to remove it.
You may want to use duct tape to cover the axle a couple layers so that if the disc hits it, it won't immediately mar it.
As I write, I though of another trick I use occasionally: Drill a whole sideways into the nut, enough to insert a hardened steel rod such as a good allen wrench. Once inserted, you can start using the lever of the rod/wrench to turn the nut. You can also use a small sharp chisel to hammer-turn the nut through the drilled hole/notch. Be sure to lubricate the axle well first.
Once done, you can easily find a regular replacement nut at your hardware store or any bike shop for very cheap. All this should cost you much less time and money compared to getting a replacement key to open it.
It's interesting the wheel (& bike) pictured is not an expensive wheel yet protected by this design.
Hope this helps!